derivative of i/(sqrt(5x^2+1))

解

\frac{d}{d x} (\frac{i}{5 x ^{2} + 1})

- \frac{5 i x}{( 1 + 5 x ^{2} ) ^{\frac{3}{2}}}

解答ステップ

\frac{d}{d x} (\frac{i}{5 x ^{2} + 1})

簡素化

\frac{i}{5 x ^{2} + 1} : i \frac{1 + 5 x ^{2}}{1 + 5 x ^{2}}

= \frac{d}{d x} (i \frac{1 + 5 x ^{2}}{1 + 5 x ^{2}})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= i \frac{d}{d x} (\frac{1 + 5 x ^{2}}{1 + 5 x ^{2}})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= i \frac{\frac{d}{d x} ( 1 + 5 x ^{2} ) ( 1 + 5 x ^{2} ) - \frac{d}{d x} ( 1 + 5 x ^{2} ) 1 + 5 x ^{2}}{( 1 + 5 x ^{2} ) ^{2}}

\frac{d}{d x} (1 + 5 x^{2}) = \frac{5 x}{1 + 5 x ^{2}}

\frac{d}{d x} (1 + 5 x^{2}) = 10 x

= i \frac{\frac{5 x}{1 + 5 x ^{2}} ( 1 + 5 x ^{2} ) - 10 x 1 + 5 x ^{2}}{( 1 + 5 x ^{2} ) ^{2}}

簡素化

i \frac{\frac{5 x}{1 + 5 x ^{2}} ( 1 + 5 x ^{2} ) - 10 x 1 + 5 x ^{2}}{( 1 + 5 x ^{2} ) ^{2}} : - \frac{5 i x}{( 1 + 5 x ^{2} ) ^{\frac{3}{2}}}

= - \frac{5 i x}{( 1 + 5 x ^{2} ) ^{\frac{3}{2}}}